Behaviors of solutions for a singular diffusion equation

This paper is devoted to some behaviors of solutions of the initial-boundary problem for a singular diffusion equation, namely, localization and large time behavior. After given some special explicit solutions it is proved that solutions of the problem possess the localization property. Next, L² decay estimate as t→∞ is proved by a rather standard energy method. Finally, by comparison with a special solution the expected L^∞ decay estimate is derived.     

Uniform and optimal estimates for solutions to singularly perturbed parabolic equations

The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L²-estimates as well as certain decay properties of bounded classical solutions to it are established. These do not depend on the value of the small perturbation parameter, and allow to prove global in time existence of strong solutions to certain boundary-value problems for ultraparabolic equations with unbounded coefficients. Optimal decay estimates are proved for such solutions. All results concerning ultraparabolic equations apply, in particular, to the Kolmogorov equation for diffusion with inertia, to the (linear) Fokker-Planck equation, to the linearized Boltzmann equation, and to some nonlinear integro-differential ultraparabolic equations of the Fokker-Planck type, arising from biophysics. Optimal decay estimates are derived for global in time strong solutions to such equations.     

Large Mass Global Solutions for a Class of L 1-Critical Nonlocal Aggregation Equations and Parabolic-Elliptic Patlak-Keller-Segel Models

We consider a class of L ¹ critical nonlocal aggregation equations with linear or nonlinear porous media-type diffusion which are characterized by a long-range interaction potential that decays faster than the Newtonian potential at infinity. The fast decay breaks the L ¹ scaling symmetry and we prove that all ‘sufficiently spread out’ initial data, even with supercritical mass, results in global, decaying solutions. In particular, we produce decaying solutions with arbitrary mass in cases for which finite time blow up solutions or non-decaying solutions are also known to exist for sufficiently large mass. This is in contrast to the classical parabolic-elliptic PKS for which essentially all solutions with supercritical mass blow up in finite time. The results with linear diffusion are proved using properties of the Fokker-Planck semi-group whereas the results with nonlinear diffusion are proved using a more interesting bootstrap argument coupling the entropy-entropy dissipation methods of the porous media equation together with higher L ^p estimates similar to those used in small-data and local theory for PKS-type equations.     

Gaussian decay of the magnetic eigenfunctions

We investigate whether the eigenfunctions of the two-dimensional magnetic Schrödinger operator have a Gaussian decay of type exp(–Cx ²) at infinity (the magnetic field is rotationally symmetric). We establish this decay if the energy (E) of the eigenfunction is below the bottom of the essential spectrum (B), and if the angular Fourier components of the external potential decay exponentially (real analyticity in the angle variable). We also demonstrate that almost the same decay is necessary. The behavior ofC in the strong field limit and in the small (B–E) limit is also studied.Partial support from the Hungarian National Foundation for Scientific Research, grant no. 1902.     

Energy decay estimates for the dissipative wave equation with space–time dependent potential

We study the asymptotic behavior of solutions of dissipative wave equations with space–time‐dependent potential. When the potential is only time‐dependent, Fourier analysis is a useful tool to derive sharp decay estimates for solutions. When the potential is only space‐dependent, a powerful technique has been developed by Todorova and Yordanov to capture the exact decay of solutions. The presence of a space–time‐dependent potential, as in our case, requires modifications of this technique. We find the energy decay and decay of the L² norm of solutions in the case of space–time‐dependent potential. Copyright © 2010 John Wiley & Sons, Ltd.     

Convergence rates to nonlinear diffusion waves for weak entropy solutions to<Emphasis Type="Italic">p</Emphasis>-system with damping

This paper studies the asymptotic behavior of weak entropy solutions to the Cauchy problem of the so-called p-system with damping. The convergence rates to nonlinear diffusion waves for weak entropy solutions are obtained in L∞norm or L² -norm. These convergence rates are the same to the decay rates of smooth solution obtained by Nishihara. They are proved by using the vanishing viscosity method and the elementary L²-energy method.     

Persistence of bounded solutions of parabolic equations under domain perturbation

The aim of this paper is to study the behavior of bounded solutions of parabolic equations on the whole real line under perturbation of the underlying domain. We give the convergence of bounded solutions of linear parabolic equations in the L ² and the L ^p -settings. For the L ^p -theory, we also prove the H?lder regularity of bounded solutions with respect to time. In addition, we study the persistence of a class of bounded solutions which decay to zero at t → ±∞ of semilinear parabolic equations under domain perturbation.     

Time decay rate for two 3D magnetohydrodynamics‐ α models

We consider the long time behavior of solutions to the magnetohydrodynamics‐ α model in three spatial dimensions. Time decay rate in L²‐norm of the solution is obtained. Similar results for a generalized Leray‐ α‐magnetohydrodynamics model are also established. As a by‐product, an optimal time decay rate for the Navier–Stokes‐ α model is achieved. Copyright © 2013 John Wiley & Sons, Ltd.     

Upper bounds of the rates of decay for solutions of the Boussinesq equations

In this paper,upper bounds of the L2-decay rate for the Boussinesq equations are considered.Using the L2 decay rate of solutions for the heat equation,and assuming that the solutions of the Boussinesq equations are smooth,we obtain the upper bounds of L2 decay rate for the smooth solutions and difference between the solutions of the Boussinesq equations and those of the heat system with the same initial data.The decay results may then be obtained by passing to the limit of approximating sequences of solutions.The main tool is the Fourier splitting method.     

On the Global Solution of a 3‐D MHD System with Initial Data near Equilibrium

In this paper, we prove the global existence of smooth solutions to the three‐dimensional incompressible magnetohydrodynamical system with initial data close enough to the equilibrium state, (e₃,0). Compared with previous works by Lin, Xu, and Zhang and by Xu and Zhang, here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is nondegenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in the earlier works. By using the Frobenius theorem and anisotropic Littlewood‐Paley theory for the Lagrangian formulation of the system, we achieve the global L¹‐in‐time Lipschitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large‐time decay rate of the solution is also obtained.© 2016 Wiley Periodicals, Inc.     

The initial value problem for a multi‐dimensional radiation hydrodynamics model with viscosity

In this paper, we study the existence and time‐asymptotic behavior of solutions to the Cauchy problem for the equations of radiation hydrodynamics with viscosity in ?³. The global existence of the solutions is obtained by using the energy method. With more elaborate energy estimates, we also give some decay rates of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

Models of thermodiffusion in 1D

The goal of the paper is to study the Cauchy problem for 1D models of thermodiffusion. We explain qualitative properties of solutions. In particular, we show which part of the model has a dominant influence on well‐posedness, propagation of singularities, L^p ? L^q decay estimates on the conjugate line, and on the diffusion phenomenon. Copyright © 2013 John Wiley & Sons, Ltd.     

Critical exponents in a degenerate parabolic equation with weighted source

This article deals with the Fujita-type theorems to the Cauchy problem of degenerate parabolic equation not in divergence form with weighted source u _t ?=?u ^p Δu?+?a(x)u ^q in ? ⁿ ?×?(0,?T), where p?≥?1, q?>?1, and the positive weight function a(x) is of the order |x| ^m with m?>??2. It was known that for the degenerate diffusion equation in divergence form, the weight function affects both of the critical Fujita exponent and the second critical exponent (describing the critical smallness of initial data required by global solutions via the decay rates of the initial data at space-infinity). Contrarily, it is interesting to prove that the weight function in the present model with degenerate diffusion not in divergence form influences the second critical exponent only, without changing the critical Fujita exponent.     

Stability of nonconstant steady‐state solutions for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor

We consider the periodic problem for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor. By choosing a suitable symmetrizers and using an induction argument on the order of the time‐space derivatives of solutions in energy estimates, we obtain the global stability of solutions with exponential decay in time near the nonconstant steady‐states for 2‐fluid nonisentropic Euler‐Poisson equations. This improves the results obtained for models with temperature diffusion terms by using the pressure functions p^ν in place of the unknown variables densities n^ν.     

Global Existence for the Vlasov-Poisson System with Steady Spatial Asymptotics

ABSTRACT

A collisionless plasma is modelled by the Vlasov-Poisson system in three space dimensions. A fixed background of positive charge—dependant upon only velocity—is assumed. The situation in which mobile negative ions balance the positive charge as | x | → ∞ is considered. Thus, the total positive charge and the total negative charge are both infinite. Smooth solutions with appropriate asymptotic behavior for large | x |, which were previously shown to exist locally in time, are continued globally. This is done by showing that the charge density decays at least as fast as | x |^?6. This article also establishes decay estimates for the electrostatic field and its derivatives.     

Large time behavior to the system of incompressible non-Newtonian fluids in R

In this paper, the authors study the large time behavior for the weak solutions to a class system of the incompressible non-Newtonian fluids in R². It is proved that the weak solutions decay in L² norm at (1+t)^−1/2 and the estimate for the decay rate is sharp in the sense that it coincides with the decay rate of a solution to the heat equation.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

Asymptotic Complexity in Filtration Equations

We show that the solutions of nonlinear diffusion equations of the form u _t = ΔΦ(u) appearing in filtration theory may present complicated asymptotics as t → ∞ whenever we alternate infinitely many times in a suitable manner the behavior of the nonlinearity Φ. Oscillatory behaviour is demonstrated for finite-mass solutions defined in the whole space when they are renormalized at each time t > 0 with respect to their own second moment, as proposed in [Tos05, CDT05]; they are measured in the L ¹ norm and also in the Euclidean Wasserstein distance W ₂. This complicated asymptotic pattern formation can be constructed in such a way that even a chaotic behavior may arise depending on the form of Φ. In the opposite direction, we prove that the assumption that the asymptotic normalized profile does not depend on time implies that Φ must be a power-law function on the appropriate range of values. In other words, the simplest asymptotic behavior implies a homogeneous nonlinearity.     

Global strong solutions and time decay of 2D tropical climate model with zero thermal diffusion

The global small solutions of the tropical climate model are obtained with the fractional dissipative terms Λ^αu in the equation of the barotropic mode u and Λ^αv in the equation of the first baroclinic mode v. More precisely, we prove for 1<α ≤ 2 that the couple system has global unique strong solutions for small initial data with critical regularities. Moreover, the smallness assumption imposed on the initial barotropic mode of the velocity can be removed if α=2. We also study the large time behavior of the constructed solutions and obtain optimal time decay rates by a pure energy argument.